Image formation in wide angle lenses

Published on by Vasili Karneichyk.

Equidistant projection (F-Theta type of lens) vs Perspective projection (F-TanTheta type of lens)

There are two primary methods of image formation in lens design. The first is perspective (F-TanTheta) projection also called Rectilinear or Orthoscopic. The second method is Equidistant (F-Theta) projection also called Equiangular.

The Perspective projection method is most often used during the design of the lenses with small and middle field of view for example in the 40- 60 degree range. This formation method maintains straight lines in images but results in space stretching. For some photography types such as satellite imaging this stretching is acceptable even at wide angles. But for many applications it is not acceptable. The photo on the right above shows a 130 degree image and the distortions.

Equidistant projection is used in wide-angle lenses like a fisheye lens. It bends straight lines but provides more than 180 degrees of lens FOV (left photo). Moreover, this projection saves angles. This is useful in astronomy photography.

There are 3 less common methods of image formation or projection, which have advantages in comparison with perspective and equidistant methods: Stereographic, Equisolid and Orthographic projections.

The key criteria used to evaluate which methods are best include:

1. Image format at given field of view depending on image construction equation;

2. Space distortion;

3. Object distortion;

4. Light distribution on image;

5. Theoretical and achievable lens field of view.

1. Image format at given field of view depending on image construction equation

1.1 Types of projections and image formation equation

Table 1.1 Image height depends on projection law

𝜔- angle of object FOV

y’ – image height

f – focal length

Next table shows how image size depends on type of projection for given angular FOV

1.2 Image size calculation of different types of projection for 120° FOV

Fig 1.2 Simplified tracing of principal ray for different projections.

Focal length 12.5 mm was previously calculated based on perspective projection according with full frame diagonal format y’=21.635 as f′=y^′/tan⁡(ω)

Then calculated focal length f’ =12.5 mm was used for all other types of projections for image height calculations.

The lens is drawn by coincided principal planes

It is seen different projections give different image height for given angular FOV (see below for details)

1.3 Correspondent standard sensor formats concerning with image height calculated for given angular FOV and equal 12.5 mm Focal Length.

Table 1.2 Correspondent standard sensor formats concerning with image height calculated for given angular FOV and equal 12.5 mm Focal Length.

2. Space distortion

2.1 Definition

Space distortion is defined as ratio of paraxial value of area s’ forming by small solid angle at image plane for given angle ω of FOV to  the area forming by equal value of solid angle at center FOV (ω=0).

Derivation of values of Space distortion was implemented in [1]

Plots of space distortion depending of type of projection are shown below.

2.2 Relative change of area value with angle of view of perspective projection

There is essential stretching space for Perspective projection after 140 deg.

2.3 Relative change of area value with angle of view of stereographic projection

There is small stretching space for Stereographic projection up to 280 deg.

2.4 Relative change of area value with angle of view of equidistant projection

There is small stretching space for Equidistant projection after 300 deg.

2.5 Relative change of square of area with angle of view for Equisolid projection

There is even space imagination for Equisolid projection.

2.6 Relative change of area value with angle of view of orthographic projection

There is compression space for Orthographic projection.

2.7 Summary

- There is stretching of image for Perspective, Stereographic and Equidistant projections.

- Distortion of Perspective projection limits FOV to 120-140 deg.

- Stereographic projections can provide maximal angular FOV 210-250 deg.

- Equidistant projections can provide maximal angular FOV 300-350 deg.

-There is compression space for  Orthographic projection.

- Orthographic projection has 180 deg. limit of angular FOV.

- There is no space distortion for Equisolid projection.

- Equisolid projection can provide maximal angular FOV up to 360 deg.

3. Object distortion

Perspective, Equidistant, Equisolid and Orthographic projections give deformation of shape of small objects through te field of view.

Only stereographic projection saves shape of small objects.

Stereographic projection preserves circles.

Stereographic projection is conformal –preserves angles of intersects of two curves

4. Distribution of illumination

4.1 Light distribution in perspective projection

Maximal field of view is limited by intensity drop-off at edge of image.

A well known formula for perspective projection describes decreasing light distribution from center to edge is

It can be used for other projections if take for regarding changing of size of square in plane of image.

Note: this equation is true if ω is equal as for object space as for image space. Image illumination can be different if the ω is different for image space what is often caused in real optical systems. This topic requires additional explaination

4.2 Distribution of illumination for different projections

Ratio of areas for Orthographic projection

Ratio of areas for Equisolid projection

For Equidistant

For Stereographic

- Orthographic projection has perfectly even illumination of image.

- Equidistant has a very slow intensity decreasing.

- Equidistant, stereographic and equi-solidprovide very wide FOV because of slow intensity drop.

5. Maximal theoretical and feasible FOV

Maximal theoretical FOV based on projection equation, acceptable space distortion and decreasing of image illumination through of imagedepends on FOV

Specific properties of particular projections

6. Examples of lens design with different image projections

Below we present examples of lens for each of projection. The lenses with close parameters for each projections were found and then additionally optimized by Zemax to match with image formation equation for each of projections.

Angular FOV is 120 deg. for all of samples

6.1 Example of lens provided perspective projection

Layout of lens with perspective (ortoscopic) projection

Distortion of the lens with perspective projection

Image simulation of the lens with perspective projection

6.3 Example of lens provided stereographic projection

layout of lens with stereographic projection

Distortion of the lens with stereographic projection

Image simulation of the lens with stereographic projection

6.4 Example of lens provided equidistant projection

Layout of lens with equidistant projection

Distortion of the lens with equidistant projection

Image simulation of the lens with equidistant projection

6.5 Example of lens provided equisolid projection

Layout of the lens with equisolid projection

Distortion of the lens with equisolid projection

Image simulation of the lens with equisolid projection

6.6 Example of lens provided orthographic projection

Layout of lens with orthographic projection

Distortion of the lens with orthographic projection

Image simulation of the lens with orthographic projection

7. Comparison of Image height and distortion (F-TanTheta in Zemax) for all types of projections

This table shows value of image height and perspective distortion (referring to perspective projection) of different types of projections

8. Conclusion

- All image formation or projection types can be useful in certain applications

- Perspective projection is useful to preserve straight lines in image. But maximal field of view should be less 140 degrees. Widely used in lenses for photography and aero photo lenses

- Stereographic projection is useful if preserving shape of small object on image plane is required. FOV can be more 180 deg. Widely used in machine vision systems.

- Equidistant projection is useful if preserving angular sizes of object on image plane is required. FOV can be more 180 deg. Widely used in fish eye lenses and astronomical cameras.

- Equisolid projection is useful if preserving constant ration of solid angles in object and image spaces is required. FOV can be more 180 deg. Used in scientific photography.

- Orthographic projection is useful if eveness of illumination throught etire image plane is required. FOV can be up to 180 degrees. Used in cheap cameras and door eye viewers.

9. References of theoretical materials

1. Field of View - Rectilinear and Fishye Lenses

http://www.bobatkins.com/photography/technical/field_of_view.html

2. Margaret M. Fleck. Perspective Projection: The Wrong Imaging Model. 1995, Technical report 95-01

http://mfleck.cs.illinois.edu/my-papers/stereographic-TR.pdf

3. Models for the various classical lens projections

http://michel.thoby.free.fr/Fisheye_history_short/Projections/Models_of_classical_projections.html

4. About the various projections of the photographic objective lenses

http://michel.thoby.free.fr/Fisheye_history_short/Projections/Various_lens_projection.html

10. References of lens design

1. U.S.Patent 3661447

2. JP #: 04-267,212

3. Imaging lens and imaging device US 20090009888 A1

4. Wide-Angle Objective. Zeiss #1058 page #0550.

5. JP Patent 4238312